Multi-spectral scattering-matrix tomography

ABSTRACT

A method for multi-spectral scattering-matrix tomography includes a step of splitting an input light signal into an incident light signal and a reference light signal. The sample light signal is directed to a sample in either a reflection configuration or a transmission configuration such that an output light signal includes light scattered from or transmitted through the sample. The incident signal and the reference light signal are directed to a camera angled to allow for amplitude and phase to be calculated by off-axis holography. A total light signal is measured with a camera that is a coherent sum of the reference light signal and the output signal. The total light signal for each light frequency and each incident angle are collected as collected total light signal data. A computing device derives an image of the sample from a calculated reflection matrix or transmission matrix or both of them.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. provisional application Ser.No. 63/270,828 filed Oct. 22, 2021, the disclosure of which is herebyincorporated in its entirety by reference herein.

TECHNICAL FIELD

In at least one aspect, the present invention relates to imagingtechniques with improved depth of field, imaging depth, resolution.

BACKGROUND

Current imaging can only image samples close to a sample's surface.Confocal microscopy is widely used for material characterization andmedical imaging. This technique has good lateral resolution but pooraxial resolution. Optical coherence tomography is currently used byophthalmologists to examine a patient's eyes. This technique, whileproviding good axial resolution, has poor lateral resolution or a highlateral resolution with a limited depth of field (trade-off betweenlateral resolution and depth of field). The imaging depth and resolutionof these methods get degrade when there are sample-induced aberrations.

Accordingly, there is a need for improved imaging techniques with acombined improved axial resolution, lateral resolution, and dept offield.

SUMMARY

In at least one aspect, a method for multi-spectral scattering-matrixtomography is provided. The method includes a step of splitting an inputlight signal into an incident light signal and a reference light signal.Characteristically, the input light signal is varied over apredetermined frequency range. The incident light signal is directed toa sample in either a reflection configuration or a transmissionconfiguration such that an output light signal includes light scatteredfrom or transmitted through the sample. The incident light signal isvaried over a predetermined range of incident angles. The output lightsignal and the reference light signal are directed to a camera such thatthe output light signal is directed at a constant angle with respect tothe reference light signal to allow for amplitude and phase to becalculated by off-axis holography. A total light signal is measured witha camera that is a coherent sum of the reference light signal and theoutput light signal. The total light signal for each light frequency andeach incident angle as collected total light signal data is collectedwith a computing device. A computing device is configured to calculate ascattering matrix and/or a reflection matrix and/or transmission matrixfrom the collected total light signal data and derive an image of thesample from the scattering matrix and/or reflection matrix and/ortransmission matrix by summing over angles and summing over lightfrequencies.

In another aspect, a multi-spectral scattering-matrix tomography systemis provided. The system includes a tunable laser that provides an inputlight signal having a light frequency varied over a predeterminedfrequency range. A beam splitter is configured to split the input lightsignal into an incident light signal and a reference light signal. Agalvanometer scanner is configured to direct the incident light signalto a sample, wherein the incident light signal is varied over apredetermined range of incident angles. A first set of opticalcomponents is configured to direct the incident light signal to thesample in either a reflection configuration or a transmissionconfiguration such that an output light signal includes light scatteredfrom or transmitted through the sample. A second set of opticalcomponents is configured to direct the output light signal and thereference light signal. Advantageously, the output light signal isdirected to the camera at a constant angle with respect to the referencelight signal to allow for amplitude and phase to be calculated byoff-axis holography. A camera is configured to measure a total lightsignal that is a coherent sum of the reference light signal and theoutput light signal. The system also includes a computing device inelectrical communication with the camera, the computing deviceconfigured to collect the total light signal for each light frequencyand each incident angle as collected total light signal data, tocalculate a scattering matrix and/or a reflection matrix and/ortransmission matrix from the collected total light signal data; and toderive an image of the sample from the scattering matrix and/or thereflection matrix and/or transmission matrix by summing over angles andsumming over light frequencies.

The foregoing summary is illustrative only and is not intended to be inany way limiting. In addition to the illustrative aspects, embodiments,and features described above, further aspects, embodiments, and featureswill become apparent by reference to the drawings and the followingdetailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

For a further understanding of the nature, objects, and advantages ofthe present disclosure, reference should be made to the followingdetailed description, read in conjunction with the following drawings,wherein like reference numerals denote like elements and wherein:

FIG. 1 . Schematic illustration of imaging inside a scattering medium.

FIG. 2 . Schematic of a multi-spectral scattering-matrix tomographysystem.

FIG. 3A. Schematic illustrating the off-axis holography method.

FIG. 3B. Map of the output light signal S_(out)'s intensity withoutbeing combined with the reference light signal S_(ref).

FIG. 3C. Map of the reference light signal S_(out)'s intensity withoutbeing combined with output light signal S_(out).

FIG. 3D. Map of the combined total signal from which amplitude and phasecan be extracted.

FIG. 4 . Maps illustrating the off-axis holography technique.

FIGS. 5-1 and 5-2 . Plots of the spectral intensity before and afteroptimization.

FIG. 6 . Plot showing a frequency-dependent phase.

FIG. 7 . Plots showing that without the synchronization, there is anunstable measurement delay.

FIG. 8 . Schematic and maps illustrating that the reference light signalcan be spatially filtered to provide a cleaner and Gaussian-like beamprofile.

FIG. 9 . Schematic illustrating the advantages of a misaligned beamsplitter.

FIG. 10 . Maps comparing confocal microscopy, OCT, and the methods setforth above are provided.

FIG. 11 . Flowchart depicting automation of the method set forth aboveis provided.

FIGS. 12-1, 12-2, 12-3, 12-4, and 12-5 . Results of performing full-wavesimulations for Maxwell's equations in 2D for a system of TiO₂nanoparticles in a tissue phantom.

FIGS. 13A, 13B, 13C, 13D, 13E, and 13F. Comparison of imagereconstruction for various techniques.

FIG. 14 . Zoomed in imaging reconstruction for the reconstructions ofFIG. 13 .

FIG. 15 . Success rates for particle identifications in constructedimages.

FIG. 16 . Axial and lateral resolution comparisons for various imagingtechniques.

FIG. 17 . Methodology for correcting for refractive index mismatchbetween air and the sample target.

FIG. 18 . Table comparing SMT and other imaging methods.

FIG. 19 . Results for examining a USAF target that is buried underneatha millimeter of mouse brain tissue.

FIG. 20 . Schematic for the arrangement of a sample for 3D imaging.

FIGS. 21A, 21B, 21C, 21D, 21E, 21F, 21G, and 21H. 3D image of the TiO₂nanoparticles for various imaging technologies.

FIG. 22 . Depth of field comparison between SMT and OCM.

FIG. 23 . Resolution of SMT.

DETAILED DESCRIPTION

Reference will now be made in detail to presently preferred embodimentsand methods of the present invention, which constitute the best modes ofpracticing the invention presently known to the inventors. The Figuresare not necessarily to scale. However, it is to be understood that thedisclosed embodiments are merely exemplary of the invention that may beembodied in various and alternative forms. Therefore, specific detailsdisclosed herein are not to be interpreted as limiting, but merely as arepresentative basis for any aspect of the invention and/or as arepresentative basis for teaching one skilled in the art to variouslyemploy the present invention.

It is also to be understood that this invention is not limited to thespecific embodiments and methods described below, as specific componentsand/or conditions may, of course, vary. Furthermore, the terminologyused herein is used only for the purpose of describing particularembodiments of the present invention and is not intended to be limitingin any way.

It must also be noted that, as used in the specification and theappended claims, the singular form “a,” “an,” and “the” comprise pluralreferents unless the context clearly indicates otherwise. For example,reference to a component in the singular is intended to comprise aplurality of components.

The term “comprising” is synonymous with “including,” “having,”“containing,” or “characterized by.” These terms are inclusive andopen-ended and do not exclude additional, unrecited elements or methodsteps.

The phrase “consisting of” excludes any element, step, or ingredient notspecified in the claim. When this phrase appears in a clause of the bodyof a claim, rather than immediately following the preamble, it limitsonly the element set forth in that clause; other elements are notexcluded from the claim as a whole.

The phrase “consisting essentially of” limits the scope of a claim tothe specified materials or steps, plus those that do not materiallyaffect the basic and novel characteristic(s) of the claimed subjectmatter.

With respect to the terms “comprising,” “consisting of,” and “consistingessentially of,” where one of these three terms is used herein, thepresently disclosed and claimed subject matter can include the use ofeither of the other two terms.

It should also be appreciated that integer ranges explicitly include allintervening integers. For example, the integer range 1-10 explicitlyincludes 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Similarly, the range 1 to100 includes 1, 2, 3, 4 . . . . 97, 98, 99, 100. Similarly, when anyrange is called for, intervening numbers that are increments of thedifference between the upper limit and the lower limit divided by 10 canbe taken as alternative upper or lower limits. For example, if the rangeis 1.1. to 2.1 the following numbers 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8,1.9, and 2.0 can be selected as lower or upper limits.

When referring to a numerical quantity, in a refinement, the term “lessthan” includes a lower non-included limit that is 5 percent of thenumber indicated after “less than.” A lower non-includes limit meansthat the numerical quantity being described is greater than the valueindicated as a lower non-included limited. For example, “less than 20”includes a lower non-included limit of 1 in a refinement. Therefore,this refinement of “less than 20” includes a range between 1 and 20. Inanother refinement, the term “less than” includes a lower non-includedlimit that is, in increasing order of preference, 20 percent, 10percent, 5 percent, 1 percent, or 0 percent of the number indicatedafter “less than.”

With respect to electrical devices, the term “connected to” means thatthe electrical components referred to as connected to are in electricalcommunication. In a refinement, “connected to” means that the electricalcomponents referred to as connected to are directly wired to each other.In another refinement, “connected to” means that the electricalcomponents communicate wirelessly or by a combination of wired andwirelessly connected components. In another refinement, “connected to”means that one or more additional electrical components are interposedbetween the electrical components referred to as connected to with anelectrical signal from an originating component being processed (e.g.,filtered, amplified, modulated, rectified, attenuated, summed,subtracted, etc.) before being received to the component connectedthereto.

The term “electrical communication” means that an electrical signal iseither directly or indirectly sent from an originating electronic deviceto a receiving electrical device. Indirect electrical communication caninvolve processing of the electrical signal, including but not limitedto, filtering of the signal, amplification of the signal, rectificationof the signal, modulation of the signal, attenuation of the signal,adding of the signal with another signal, subtracting the signal fromanother signal, subtracting another signal from the signal, and thelike. Electrical communication can be accomplished with wiredcomponents, wirelessly connected components, or a combination thereof.

The term “one or more” means “at least one” and the term “at least one”means “one or more.” The terms “one or more” and “at least one” include“plurality” as a subset.

The term “substantially,” “generally,” or “about” may be used herein todescribe disclosed or claimed embodiments. The term “substantially” maymodify a value or relative characteristic disclosed or claimed in thepresent disclosure. In such instances, “substantially” may signify thatthe value or relative characteristic it modifies is within ±0%, 0.1%,0.5%, 1%, 2%, 3%, 4%, 5% or 10%.

The term “computing device” refers generally to any device that canperform at least one function, including communicating with anothercomputing device. In a refinement, a computing device includes a centralprocessing unit that can execute program steps and memory for storingdata and a program code. Examples of computing devices include, but arenot limited to, desktop computers, notebook computers, laptop computers,mainframes, mobile phones, headsets such as augmented reality headsets,virtual reality headsets, mixed reality headsets, augmented realitydevices, virtual reality devices, mixed reality devices, and the like.

When a computing device is described as performing an action or methodstep, it is understood that the one or more computing devices areoperable to and/or configured to perform the action or method steptypically by executing one or more lines of source code. The actions ormethod steps can be encoded onto non-transitory memory (e.g., harddrives, optical drive, flash drives, and the like).

The processes, methods, or algorithms disclosed herein can bedeliverable to/implemented by a processing device, controller, orcomputer, which can include any existing programmable electronic controlunit or dedicated electronic control unit. Similarly, the processes,methods, or algorithms can be stored as data and instructions executableby a controller or computer in many forms including, but not limited to,information permanently stored on non-writable storage media such as ROMdevices and information alterably stored on writeable storage media suchas floppy disks, magnetic tapes, CDs, RAM devices, and other magneticand optical media. The processes, methods, or algorithms can also beimplemented in a software executable object. Alternatively, theprocesses, methods, or algorithms can be embodied in whole or in partusing suitable hardware components, such as Application SpecificIntegrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs),state machines, controllers or other hardware components or devices, ora combination of hardware, software and firmware components.

Abbreviations

“CCD” means charged coupled device.

“DOF” means depth of field.

“FWHM” means full-width half-maximum.

“HWP” means half-wave plate.

“ISAM” means interferometric synthetic aperture microscopy.

“NA” means numerical aperture.

“NUFFT” means non-uniform fast Fourier transform.

“OCM” means optical coherence microscopy.

“OCT” means optical coherence tomography.

“PBS” means polarizing beam splitter.

“SMT” means scattering matrix tomography.

Referring to FIG. 1 , a schematic illustration of imaging inside ascattering medium is provided. The figure shows a medium with large andsmall scattering centers dispersed in a medium. In many applications,the medium needs to be probed from the outside in a non-invasive manner.The multi-spectral scattering-matrix tomography systems set forth hereinallow for improved imaging of the inside the scattering medium forapplications such as biomedical imaging and non-destructive devicetesting.

Referring to FIG. 2 , a schematic of a multi-spectral scattering-matrixtomography system is provided. Multi-spectral scattering-matrixtomography system includes a tunable laser 12 (e.g., a Ti-sapphirelaser) that provides an input light signal S_(in) having a lightwavelength (and therefore frequency) varied over a predeterminedwavelength range (and therefore a predetermined frequency range). In arefinement, the predetermined wavelength range is from 700 nm to 1000 nm(i.e., 430 THz to 300 THz). Tunable laser 12 is in electricalcommunication with computing device 14. Wavemeter 16 is also incommunication with computing device 14 to monitor the frequency of theinput light signal. A beam splitter 20 is configured to split the inputlight signal S_(in) into an incident light signal S_(sam) (sometimesreferred to as a sample light signal) and a reference light signalS_(ref). A galvanometer scanner 24 is configured to direct the incidentlight signal to a sample such that the incident light signal is variedover a predetermined range of incident angles.

A first set of optical components 28 is configured to direct theincident light signal to the sample in either a reflection configurationor a transmission configuration such that an output light signalincludes light scattered from or transmitted through the sample. Forexample, the incident light signal is passed through scan lens 30 andthen through tube lens 32. Flip mirror 34 is used to determine if system10 operates in the reflection configuration or the transmissionconfiguration. When the flip mirror 34 is not in place, path Pr isfollowed, and the reflection configuration is selected. The incidentlight signal passes from the flip mirror to mirrors 36 and 38 throughlens 40 and then to beam splitter 42. From beam splitter 42, theincident light signal is directed through objective 44 to sample 46.When the flip mirror 34 is in place, path Pt is followed, and thetransmission configuration is selected. For example, the incident lightsignal passes from the flip mirror 34 to mirror 50 and then through tubelens 52. The incident light signal is then directed by mirror 54 throughobjective 56 and finally to sample 46.

Still referring to FIG. 2 , a second set of optical components 57 isconfigured to direct the output light signal and the reference lightsignal to high-speed camera 58. The output light signal is directed tothe camera at a constant angle with respect to the reference lightsignal to allow for amplitude and phase to be calculated by off-axisholography. For example, the reference light signal passes throughhalf-wave plate 60 and delay line 62 and then to mirrors 64, 66, 68, and70. The reference light signal then passes through beam expander 72 andlens 74. The reference light signal passes through spatial filter 76onto beam splitter 42 where it coherently adds to the output lightsignal from the sample to form a total light signal. In a refinement,galvanometer scanner 24 and camera 58 are synchronized with a triggersignal.

Still referring to FIG. 2 , high-speed camera 58 (e.g., a CCD camera) isconfigured to measure a total light signal that is a coherent sum of thereference light signal and the output signal. Computing device 14 is inelectrical communication with the camera. The computing device isconfigured to collect the total light signal for each light frequencyand each incident angle as collected total light signal data, tocalculate a scattering matrix and/or a reflection matrix and/or atransmission matrix from the collected total light signal data, and toderive an image of the sample from the scattering matrix and/or thereflection matrix and/or the transmission matrix by summing over anglesand summing over light frequencies.

Still referring to FIG. 2 , computing device 14 is configured tocalculate the scattering matrix and/or the reflection matrix and/or thetransmission matrix is determined by Fourier transforming the collectedtotal light signal data to form a transformed collected total signaldata and performing an inverse Fourier transform on a first-order regionof the transformed collected total signal data to determine amplitudeand phase of the output signal. In this context, the term “scatteringmatrix” refers to the matrix that is multiplied by a vector formed bythe weights of a plurality of incident plane waves that whensuperimposed form the incident light signal (i.e., the input) to providea vector formed by the weights of a plurality of scattered plane wavesthat when superimposed form the scattered light signal (i.e., theoutput) where the scattered light signal is the light scattered from asample. When only the scattered light from the opposite side of thesample is considered, the scattering matrix is referred to as thetransmission matrix. When only the scattered light from the same side ofthe sample is considered, the scattering matrix is referred to as thereflection matrix. The scattering matrix can be measured non-invasivelyas shown below. Once the scattering matrix or reflection matrix isdetermined, the response of a system can be synthesized given anarbitrary input light signal. Advantageously, this allows the ability todigitally achieve perfect spatio-temporal focusing with both inputspatial gating, output spatial gating, and time gating. With respect tothe incident light signal, a superposition of plane waves acrossdifferent incident angles can be used to focus the incident light signalat a predetermined position r₀ to provide input spatial gating as shownby the following equation:

$\sum\limits_{k_{in}}e^{{{ik}_{in} \cdot {({r - r_{0}})}} - {i\omega t}}$

Similarly, time gating is obtained when the summation is overfrequencies. For time gating, a pulse is obtained that arrives atposition r₀ at time t equal to 0 as shown in the following equation:

$\sum\limits_{\omega}e^{{{ik}_{in} \cdot {({r - r_{0}})}} - {i\omega t}}$

Therefore, if summations are performed over both k_(in) and ω, aspatio-temporal focusing input is obtained as follows:

$\sum\limits_{\omega}{\sum\limits_{k_{in}}e^{{{ik}_{in} \cdot {({r - r_{0}})}} - {i\omega t}}}$

Alternatively, this can be expressed as an integral over frequencies was follows:

$\int{d\omega{\sum\limits_{\omega}e^{{{ik}_{in} \cdot {({r - r_{0}})}} - {i\omega t}}}}$

This results in the incident light signal being focused at r₀ at time tequal to 0.

In a variation, the hyper-spectral reflection matrix is measured. Thereflection matrix can provide the complex amplitude of reflection todifferent outgoing angles given different incident angle directionsacross different frequencies. Once this data is obtained, the incidentplane wave can be digitally synthesized to be focused at a predeterminedposition r₀ as shown above by multiplying by the reflection matrix togive the reflective light to different outgoing directions. One can thensum over the different outgoing waves and evaluate the outgoing wave atthe same position r₀. This is where the response is maximized if we havescattering going on at this position. This procedure gives the outputspatial gating. To do a time gating, a summation over frequency isperformed as described above to provide the response at time t=0.Overall, this triple summation gives us a response that will bemaximized when there is a target at position r₀. Therefore, squaring theresponse gives us a real intensity that can be scanned across thepositions r₀ giving 3D volumetric image. This combination of spatialgating and time gating can be performed simultaneously across the entirevolume. Advantageously, there is no longer any tradeoff between depth offocus and lateral resolution. This triple summation is described by thefollowing formula:

${I_{SMT}\left( r_{0} \right)} = {❘{\sum\limits_{\omega}{\sum\limits_{k_{out}}{\sum\limits_{k_{in}}{e^{{i({k_{out} - k_{in}})} \cdot r_{0}}{S\left( {\omega,k_{out},k_{in}} \right)}}}}}❘}^{2}$

where:I_(SMT) is the image intensity as a function of position r₀ in thesample;r₀ is a position vector of a point in the sample;S(ω, k_(out), k_(in)) is the element of the scattering matrix for theincidence channel with k_(in) and the reflection channel with k_(out);k_(in) is the wavevector of the incident light signal;k_(out) is the wavevector of the output (i.e., reflected) light signal;andω is the light frequency. Alternatively, the image intensity is foundfrom the following formula when system 10 is in the reflectionconfiguration where the integral over ω is approximated by a summationover ω:

${I_{SMT}\left( r_{0} \right)} = {❘{\sum\limits_{\omega}{\sum\limits_{k_{out}}{\sum\limits_{k_{in}}{e^{{i({k_{out} - k_{in}})} \cdot r_{0}}{R\left( {\omega,k_{out},k_{in}} \right)}}}}}❘}^{2}$

where:I_(SMT) is the image intensity as a function of position r₀ in thesample;r₀ is a position vector of a point in the sample;R(ω, k_(out), k_(in)) is the element of the reflection matrix for theincidence channel with k_(in) and the reflection channel with k_(out);k_(in) is the wavevector of the incident light signal;k_(out) is the wavevector of the output (i.e., reflected) light signal;andω is the light frequency.

In a variation, the image intensity is found from the following formulawhen system 10 is in the transmission configuration where the integralover ω is approximated by a summation over ω:

${I_{SMT}\left( r_{0} \right)} = {❘{\sum\limits_{\omega}{\sum\limits_{k_{out}}{\sum\limits_{k_{in}}{e^{{i({k_{out} - k_{in}})} \cdot r_{0}}{T\left( {\omega,k_{out},k_{in}} \right)}}}}}❘}^{2}$

where:I_(SMT) is the image intensity as a function of position r₀ in thesample;r₀ is a position vector of a point in the sample;T(ω, k_(out), k_(in)) is the element of the transmission matrix for theincidence channel with k_(in) and the reflection channel with k_(out);k_(in) is the wavevector of the incident light signal;k_(out) is the wavevector of the output (i.e., reflected) light signal;andω is the light frequency.

In a refinement, computing device 14 is configured to determine an imageintensity from the following equation when system 10 is in thereflection configuration:

${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}r_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$

where:I(r) is the image intensity as a function of position r in the sample;r is a position vector of a point in the sample;r_(ba) is the element of the reflection matrix for the a-th incidenceand b-th reflection channel;a is a label for an angle of incidence;b is a label for an angle of reflection;k_(a) is the wavevector of the incident light signal;k_(b) is the wavevector of the output (i.e., reflected) light signal;andω is the light frequency. It should be noted that the image intensity isexpressible as:

I(r)∝|∫dω(A _(r) ^(Born))†R| ²,

where R is the reflection matrix and A_(r) ^(Born) is the Born matrix.Therefore, I(r) is calculated in the context of the Born Approximationwhere the multiple scattering events are ignored. However, the summingover frequencies and angle of incidence allows for multiple-scatteringcontributions add in quasi-random phase, thereby canceling. AttachedExhibit A provides a derivation of A_(r) ^(Born). Exhibit A is part ofthe specification and incorporated herein in its entirety.

In another refinement, image intensity is determined from the followingequation when system 10 is in the transmission configuration:

${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}t_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$

where:I(r) is the image intensity as a function of position r in the sample;r is a position vector of a point in the sample;t_(ba) is the element of the transmission matrix for the a-th incidenceand b-th transmitted channel;a is a label for an angle of incidence;b is a label for an angle of transmitted light;k_(a) is the wavevector of the incident light signal;k_(b) is the wavevector of the output light signal; andω is the light frequency.

In summary, the components of SMT summation over incident angles provideinput spatial gating, summation over output angle gives us outputspatial gating, and summation over frequency gives time gating. Itshould be appreciated that the measured reflection matrix is not focusedeither in time or in space. However, all of this refocusing is performeddigitally during post-processing. When the triple summation is performedto give an image, the single scattering signals from the targets add upin phase. Meanwhile, the multiple scattered light signal do not add upin phase in the summations and, therefore, are attenuated (e.g.,rejected). In addition to this threefold gating, SMT also allowscorrection for various aberrations digitally. For example, there istypically an index change going from air to the sample target whichdegrades the focusing quality. This can be corrected for by choosingmomentum k_(in) and k_(out) by using the momentum inside the sample. Inanother example, there are often aberrations and dispersion from theoptical system. These can be corrected for by measuring the reflectionmatrix of a mirror which would be one if there is no system aberration.Therefore, minus the phase of the mirror reflection matrix providescorrection for both the chromatic and spatial aberrations of the opticalelements. In another example, there can also be dispersion from thesample which can be reduced by optimizing spectral phase in thefrequency summation. Finally, there can also be spatial aberrationsintroduced by the sample target. Similarly, these latter aberrations canbe corrected for by optimizing the “in-coming”-angle- and the“outgoing”-angle-dependent phases during the angular summations.Therefore, scattering matrix tomography provides not onlyspatio-temporal gating across whole volume but a comprehensivecorrection for various aberrations that can be done digitally.

FIGS. 3A-D and 4 depict the operation of the off-axis holographtechnique that is implemented by a multi-spectral scattering-matrixtomography system 10. FIG. 3A schematically depicts an off-axisholograph technique in which the output light signal S_(out) is directedto camera 58 at a constant angle with respect to the reference lightsignal S_(ref) to allow for amplitude and phase to be calculated byoff-axis holography. FIG. 3B depicts a map of the output light signalS_(out)'s intensity without being combined with the reference lightsignal S_(ref) while FIG. 3C depicts a map of the reference light signalS_(ref)'s intensity without being combined with output light signalS_(out). FIG. 3D provides a map of the combined total signal intensityfrom which amplitude and phase can be extracted.

FIG. 4 provides plots illustrating the extraction of amplitude and phaseinformation from the combined total light signal (S_(out)+S_(ref)). Inthis method, the collected total light signal data for a single incidentangle is Fourier transformed to form a transformed collected totalsignal data. An inverse Fourier transform is then performed on afirst-order (e.g., −1 or 1 order) region (i.e., by cropping the Fourierspace) of the transformed collected total signal data to determine theamplitude and phase of the output signal. This method allows row ofvalues of the reflection matrix (or transmission matrix) to becalculated. The entries for each row is the k_(b) vector (e.g. kx and kyfor each output channel b) for the output channel b. This reflectionmatrix (or transmission matrix) is constructed by scanning angles ofincidence.

Referring to FIGS. 2 and 5 , the effects of the variation of input lightsignal intensity of the varying frequencies are illustrated. FIG. 5provides plots of the spectral intensity before and after optimization.As set forth above, the input light signal S_(in) is modified over thepredetermined frequency range. Therefore, system 10 can include lightattenuator 80 configured to modify input light signal intensity over thepredetermined frequency range to minimize light intensity differences.

Referring to FIGS. 2 and 6 , the effects of the frequency-dependentphase are illustrated. To mitigate the effect of the frequency-dependentphase, system 10 can include dispersion compensator 82 is the referencelight signal path.

Referring to FIGS. 2 and 7 , the effect for synchronizing thegalvanometer scanner and the camera are illustrated. As set forth above,galvanometer scanner 24 and the camera 58 are synchronized with atrigger signal from one of the devices to the other. FIG. 7 shows thatwithout the synchronization, there is an unstable delay of up to 1 msbetween different measurements. With synchronization, the measurementsare repeatable.

Referring to FIGS. 2 and 8 , the effects of filtering the referencelight signal are provided. As set forth above, system 10 includesspatial filter 76 configured to spatially filter the reference lightsignal to provide a cleaner and Gaussian-like beam profile. FIG. 8confirms that the reference light signal can be spatially filtered toprovide a cleaner and Gaussian-like beam profile.

Referring to FIGS. 2 and 9 , the advantages of a non-45 degree beamsplitter are provided. The non-45 degree alignment of beam splitter 42allows for spurious light signals from the beam split to avoid camera58.

Referring to FIG. 10 , maps comparing conformal microscopy, OCT, and themethods set forth above are provided. The multi-spectralscattering-matrix tomography method set forth herein provides improvedlateral resolution (0.4λ/NA) and improved axial resolution (0.44λ²/Δλ)

Referring to FIG. 11 , a flowchart depicting automation of the methodset forth above are provided. In step 100, the SMT system isinitialized. In decision box 102, a determination is made if all thedesired wavelengths of light have been scanned. If the entirepredetermined wavelength range has been scanned, the method stops. Ifthe entire predetermined wavelength range has not been scanned the laserlight source and attenuator are tuned to a given wavelength.

FIG. 12 provides the results of performing full-wave simulations forMaxwell's equations in 2D for a system of TiO₂ nanoparticles in a tissuephantom. The intensity profile on the left is for incident lightcontacting imaging target with an angle of incidence of about 15°. FIG.12-2 and FIG. 12-3 show the ground truth in the pink window near thefront of the sample and the green window deeper into the sample. The reddots are TiO2 particles that we want to image, which are buried in thetissue phantom consisting of many larger lower index particles. FIG.12-4 and FIG. 12-5 provide the intensity profiles in these two windows.It is observed that near the sample front there is visiblecorrespondence between the scatter location and the high intensityprofile, but deeper into the sample there is no longer such visible. Inthe full-wave simulation, we get the reflected waves and project theminto different reflected angles to mimic experimentally measurement andthat gives us one column of the reflection matrix. Conventionally, onewould have to perform these simulations repeatedly across hundreds ofincident angles to build the reflection matrix. However, we havedeveloped an efficient software called MESTI (Maxwell's Equations Solverwith Thousands of Inputs) which is open source of Gitbhub which cancompute the whole scattering matrix simultaneously efficiently(https://github.com/complexphoton/mesti.m). Using this software, one canpredict reflection matrix R(ω, k_(out), k_(in)) of this 400 micron by600 micron system across 450 wavelengths (700 to 1000 nm) at 0.5numerical aperture (about 600˜900 angles per wavelength).

FIGS. 13A, 13B, 13C, 13D, 13E, and 13F provide a comparison of imagereconstruction for various techniques. FIG. 13A provides the groundtruth showing the location of TiO₂ nanoparticles and FIG. 13A we get byevaluating this equation for scattering matrix tomography. These figuresshow that locations of the TiO₂ nanoparticles can be faithfullyreconstructed with high resolution even though only data in thereflection in the far field is used. In SMT, a coherent summation of thefrequencies is performed. As a comparison in FIG. 13C, an incoherentsummation of frequencies is performed by summing over the intensities.This mimics confocal microscopy with the confocal spatial gating in theinput and output. However, there is no temporal gating. It is observedthat axial resolution is reduced because of the lack of time gating.

FIGS. 13D, 13E, and 13F provide simulations that mimic what happens inthe OCT, OCM, and ISAM. In the OCT experiment, the numerical aperture islow such that the depth of field is large. Here the lateral resolutionis reduced and also imaging depth is reduced. At the focal plane OCM canhave high resolution but the resolution degrades away from the focalplane and again, doesn't image as deep. ISAM can improve the lateralresolution near the surface of the sample because scattering is weakthere. However, ISAM's performance is still limited at other locationsin the sample target. This is because in ISAM the output always dependson the input so that its gating efficiency decreases away from the focalplane. FIG. 14 provides zoomed in imaging reconstruction for thereconstructions of FIG. 13 . At the front part of the sample, SMT givesa high resolution image. In contrast, broadband confocal has a loweraxial resolution and OCT has a low lateral resolution. It is observedthat several targets missed by OCT while features of OCT that do notcorrespond to a real target are observed in the images. OCM and ISAM donot work very well at the front part of the sample because they are faraway from their focal plane. Deeper inside the target sample, SMT canstill reconstruct images with high resolution and deeper depths incontrast to the other imaging methods.

FIG. 15 provides success rates for particle identifications inreconstructed images. Since the ground nanoparticle locations for thesimulations are known, the success rate of the images in identifying theparticle locations can be determined. If the imaging depth is defined aswhere the success rate maintains above 50%, it is observed that SMTachieves a depth of more than 2× deeper than OCT.

FIG. 16 provides axial and lateral resolution comparisons for variousimaging techniques. SMT is observed to provide very good sub-micronaxial and lateral resolution across the entire volume.

FIG. 17 depict methodology for correcting for refractive index mismatchbetween air and the sample target. In this regard, the interface betweenair and the sample target affects focusing by a typical objective inthat the focus shifts light from different angles resulting in a focusto different points which degrades the focusing quality. This can becorrected digitally in SMT by using use the momentum in the mediumitself. This effectively creates the input and output wave frontperfectly focused to a single point for all different angles acrossevery depth that is scanned.

FIG. 18 provides a table comparing the SMT and other imaging methods.SMT can image deeper than the other methods maintaining high lateral andaxial resolutions across a large volume by digital scanning whilecorrecting index mismatch between a target and air. In the simulations,the tissue phantom is homogeneous in a sense that its effective index isroughly the same across the volume. However, in real biological tissue,there will be strong spatially inhomogeneities that also give rise tospatial aberration which can be digitally corrected.

FIG. 19 provides experimental results for examining a USAF target thatis buried underneath a millimeter of mouse brain tissue. FIG. 19provides a comparison between broadband confocal, OCT, OCM, and SMT. Inthe broadband confocal method, the summation over frequencies isincoherent whereas in the SMT method the summation is coherent.Therefore, in the broadband confocal method time gating is ineffectivein suppressing multiple scattered light resulting in no image at thedepths of the USAF target. With OCT (NA=0.1), a better gating rejectionof multiple scattered light is observed but with low spatial gating andlow resolution. When the NA is increased to 0.5 we have OCM then we havebetter spatial gating now we start to some feature of the USAF targetbut we still cannot see the structure. The SMT results show asignificant improvement in the image clarity. The SMT images have beencorrected for spatial and chromatic aberration of the optical system bycalibrating against a mirror. Sample's dispersion was corrected for byoptimizing the spectral phase. The SMT images have also corrected forindex mismatch correction as described above. Finally, optimization overthe angle-dependent phase corrects for spatial aberration of mouse braintissue.

FIGS. 20, 21A, 21B, 21C, 21D, 21E, 21F, and 21G demonstrate theapplicability of SMT to 3D imaging. FIG. 20 provides a schematic for thearrangement of a sample for 3D imaging. FIG. 21 provide 3D image of theTiO₂ nanoparticles for various imaging technologies. SMT provides clearimages while broad band confocal shows poor resolution with barely anyfeatures being observed. OCT provides low lateral resolution while OCMonly shows nanoparticles near the focal plane.

FIG. 22 provides a depth of field (DOF) comparison between SMT and OCM.The depth of field is defined as when the lateral resolution degrades bya factor of V. For OCM, the DOF is twice the Rayleigh length. Incontrast to SMT, OCM does not have good spatial focus outside of itsdepth of field. FIG. 23 provides plots for axial and lateral resolutionat different sample locations.

While exemplary embodiments are described above, it is not intended thatthese embodiments describe all possible forms of the invention. Rather,the words used in the specification are words of description rather thanlimitation, and it is understood that various changes may be madewithout departing from the spirit and scope of the invention.Additionally, the features of various implementing embodiments may becombined to form further embodiments of the invention.

What is claimed is:
 1. A method for multi-spectral scattering-matrixtomography comprising a) splitting an input light signal into anincident light signal and a reference light signal, wherein the inputlight signal is varied over a predetermined frequency range; b)directing the incident light signal to a sample in either a reflectionconfiguration or a transmission configuration such that an output lightsignal includes light scattered from or transmitted through the sample,wherein the incident light signal is varied over a predetermined rangeof incident angles; c) directing the output light signal and thereference light signal to a camera, the output light signal directed ata constant angle with respect to the reference light signal to allow foramplitude and phase to be calculated by off-axis holography; d)measuring with the camera a total light signal that is a coherent sum ofthe reference light signal and the output light signal; e) collectingthe total light signal for each light frequency and each incident angleas collected total light signal data; f) calculating with a computingdevice a scattering matrix or a reflection matrix or a transmissionmatrix from the collected total light signal data; and g) deriving animage of the sample from the scattering matrix or reflection matrix ortransmission matrix by summing over angles and summing over lightfrequencies.
 2. The method of claim 1, wherein the scattering matrix,the reflection matrix or the transmission matrix is determined byFourier transforming the collected total light signal data to form atransformed collected total signal data and performing an inverseFourier transform on a first-order region of the transformed collectedtotal signal data to determine amplitude and phase of the output lightsignal.
 3. The method of claim 1 wherein an image intensity isdetermined from:${I_{SMT}\left( r_{0} \right)} = {❘{\sum\limits_{\omega}{\sum\limits_{k_{out}}{\sum\limits_{k_{in}}{e^{{i({k_{out} - k_{in}})} \cdot r_{0}}{S\left( {\omega,k_{out},k_{in}} \right)}}}}}❘}^{2}$where: I_(SMT) is the image intensity as a function of position r₀ inthe sample; r₀ is a position vector of a point in the sample; S(ω,k_(out), k_(in)) is the element of the scattering matrix for theincidence channel with k_(in) and the reflection channel with k_(out);k_(in) is the wavevector of the incident light signal; k_(out) is thewavevector of the output (i.e., reflected) light signal; and ω is thelight frequency.
 4. The method of claim 1 wherein an image intensity isdetermined from:${I_{SMT}\left( r_{0} \right)} = {❘{\sum\limits_{\omega}{\sum\limits_{k_{out}}{\sum\limits_{k_{in}}{e^{{i({k_{out} - k_{in}})} \cdot r_{0}}{R\left( {\omega,k_{out},k_{in}} \right)}}}}}❘}^{2}$where: I_(SMT) is the image intensity as a function of position r₀ inthe sample; r₀ is a position vector of a point in the sample; R(ω,k_(out), k_(in)) is the element of the reflection matrix for theincidence channel with k_(in) and the reflection channel with k_(out);k_(in) is the wavevector of the incident light signal; k_(out) is thewavevector of the output (i.e., reflected) light signal; and ω is thelight frequency.
 5. The method of claim 1 wherein an image intensity isdetermined from:${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}r_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$where: I(r) is the image intensity as a function of position r in thesample; r is a position vector of a point in the sample; r_(ba) is anelement of the reflection matrix for an a-th incidence and b-threflection channel; a is a label for a channel of incidence; b is alabel for a channel of reflection; k_(a) is the wavevector of theincident light signal; k_(b) is the wavevector of the output lightsignal; and ω is the light frequency.
 6. The method of claim 5 whereinthe image intensity is expressible as:I(r)∝|∫dω(A _(r) ^(Born))†r′| ².
 7. The method of claim 1 wherein animage intensity is determined from:${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}t_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$where: I(r) is the image intensity as a function of position r in thesample; r is a position vector of a point in the sample; t_(ba) is anelement of the transmission matrix for an a-th incidence and b-thtransmitted channel; a is a label for an angle of incidence; b is alabel for an angle of transmitted light; k_(a) is the wavevector of theincident light signal; k_(b) is the wavevector of the output lightsignal; and ω is the light frequency.
 8. The method of claim 1 furthercomprising modifying input light signal intensity over the predeterminedfrequency range to minimize light intensity differences.
 9. The methodof claim 8, wherein the input light signal intensity is modified with alight attenuator.
 10. The method of claim 1, wherein the incident lightsignal is varied over the predetermined range of incident angles with agalvanometer scanner.
 11. The method of claim 10, wherein thegalvanometer scanner and the camera are synchronized with a triggersignal.
 12. The method of claim 1, wherein the reference light signal isspatially filtered to provide a cleaner and Gaussian-like beam profile.13. A multi-spectral scattering-matrix tomography system comprising: atunable laser providing an input light signal having a light frequencyvaried over a predetermined frequency range; a beam splitter configuredto split the input light signal into an incident light signal and areference light signal; a galvanometer scanner configured to direct theincident light signal to a sample, wherein the incident light signal isvaried over a predetermined range of incident angles; a first set ofoptical components configured to direct the incident light signal to thesample in either a reflection configuration or a transmissionconfiguration such that an output light signal includes light scatteredfrom or transmitted through the sample; a camera configured to measure atotal light signal that is a coherent sum of the reference light signaland the output light signal; a second set of optical componentsconfigured to direct the output light signal and the reference lightsignal to the camera, the output light signal being directed at aconstant angle with respect to the reference light signal to allow foramplitude and phase to be calculated by off-axis holography; a computingdevice in electrical communication with the camera, the computing deviceconfigured to collect the total light signal for each light frequencyand each incident angle as collected total light signal data, tocalculate a scattering matrix or a reflection matrix or a transmissionmatrix from the collected total light signal data; and to derive animage of the sample from the reflection matrix or transmission matrix bysumming over angles and summing over light frequencies.
 14. The systemof claim 13, wherein the computing device is further configured todetermine the scattering matrix or the reflection matrix or thetransmission matrix by Fourier transforming the collected total lightsignal data to form a transformed collected total signal data andperforming an inverse Fourier transform on a first-order region of thetransformed collected total signal data to determine amplitude and phaseof the output light signal.
 15. The system of claim 14, wherein thecomputing device is further configured to determine an image intensityfrom:${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}r_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$where: I(r) is the image intensity as a function of position r in thesample; r is a position vector of a point in the sample; r_(ba) is anelement of the reflection matrix for an a-th incidence and b-threflection channel; a is a label for an angle of incidence; b is a labelfor an angle of reflection; k_(a) is the wavevector of the incidentlight signal; k_(b) is the wavevector of the output light signal; and ωis the light frequency.
 16. The system of claim 14, wherein thecomputing device is further configured to determine an image intensityfrom:${I(r)} \propto {❘{\int{d\omega{\sum\limits_{b,a}{e^{{ik}_{b} \cdot r}t_{ba}e^{- {{ik}_{a} \cdot r}}}}}}❘}^{2}$where: I(r) is an image intensity as a function of position r in thesample; r is a position vector of a point in the sample; t_(ba) is anelement of the transmission matrix for an a-th incidence and b-thtransmitted channel; a is a label for an angle of incidence; b is alabel for an angle of transmitted light; k_(a) is the wavevector of theincident light signal; k_(b) is the wavevector of the output lightsignal; and ω is the light frequency.
 17. The system of claim 13,further comprising a light attenuator configured to modify input lightsignal intensity over the predetermined frequency range to minimizelight intensity differences.
 18. The system of claim 13, wherein thegalvanometer scanner and the camera are synchronized with a triggersignal.
 19. The system of claim 13, further comprising a spatial filterconfigured to spatially filter the reference light signal to provide acleaner and Gaussian-like beam profile.